| L(s) = 1 | − 3.01i·3-s + 16.1·5-s − 66.9·7-s + 71.9·9-s − 221. i·11-s + 65.3i·13-s − 48.6i·15-s − 59.5i·17-s − 328.·19-s + 201. i·21-s − 623. i·23-s − 364.·25-s − 460. i·27-s − 1.27e3i·29-s + (328. − 903. i)31-s + ⋯ |
| L(s) = 1 | − 0.334i·3-s + 0.646·5-s − 1.36·7-s + 0.888·9-s − 1.83i·11-s + 0.386i·13-s − 0.216i·15-s − 0.206i·17-s − 0.910·19-s + 0.457i·21-s − 1.17i·23-s − 0.582·25-s − 0.631i·27-s − 1.51i·29-s + (0.341 − 0.939i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.781427 - 1.11516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.781427 - 1.11516i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-328. + 903. i)T \) |
| good | 3 | \( 1 + 3.01iT - 81T^{2} \) |
| 5 | \( 1 - 16.1T + 625T^{2} \) |
| 7 | \( 1 + 66.9T + 2.40e3T^{2} \) |
| 11 | \( 1 + 221. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 65.3iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 59.5iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 328.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 623. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.27e3iT - 7.07e5T^{2} \) |
| 37 | \( 1 + 477. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 329.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 3.32e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 742.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.74e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.72e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 4.98e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 6.59e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 8.23e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 7.16e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 6.39e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.78e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.27e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 3.40e3T + 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.70130421694488754182772063522, −11.34666703066647042085509454278, −10.14876600009176273825800424424, −9.350686111243821539353118003985, −8.095259267828933819212101408464, −6.50108429434255454836686890227, −6.04969757585370498707350593451, −4.03950673462484467389761341631, −2.52172281367231965411763856501, −0.56317554929320268358147586530,
1.87258333600320374393363969336, 3.62293359438886084606599589571, 5.01167212139019444327514336534, 6.48456138695687252707373258304, 7.34691640449658091402833318497, 9.191121805336042625425424560918, 9.899334617821289119853787730511, 10.51064537594707018161952616632, 12.47214384268604914677863361037, 12.74724347311279031905550348191
